On $L-$generalized filters and nets
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In this paper, we consider the $L-$generalized filters and $L-$generalized nets on the non empty universal set $X$ to be a natural generalization of the crisp generalized filters and the crisp generalized nets respectively. In each the crisp and $L-$generalized cases, the extended and restricted filters (nets) will be obtained respectively. The construction of convergence to limit points of the crisp generalized and $L-$generalized filters (nets) will be studied respectively. We also show that any crisp generalized net on the set $X$ associates a unique $L-$generalized net on $X,$ and in general the converse is not true. In addition to that we also show that for any given $L-$generalized filter there exists the $L-$generalized net, and the vice-versa is true. Finally, the relations between the convergence of the $L-$generalized filters and the $L-$generalized nets will be outlined. Moreover, to support these concepts and relations, the construction of some examples in all sections of paper will be studied.
Abstract
In this paper, we consider the $L-$generalized filters and $L-$generalized nets on the non empty universal set $X$ to be a natural generalization of the crisp generalized filters and the crisp generalized nets respectively. In each the crisp and $L-$generalized cases, the extended and restricted filters (nets) will be obtained respectively. The construction of convergence to limit points of the crisp generalized and $L-$generalized filters (nets) will be studied respectively. We also show that any crisp generalized net on the set $X$ associates a unique $L-$generalized net on $X,$ and in general the converse is not true. In addition to that we also show that for any given $L-$generalized filter there exists the $L-$generalized net, and the vice-versa is true. Finally, the relations between the convergence of the $L-$generalized filters and the $L-$generalized nets will be outlined. Moreover, to support these concepts and relations, the construction of some examples in all sections of paper will be studied.
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